Running head: CRYPTOGRAPHY 1

Cryptography: Mathematical Advancements on Cyber Security

Kristin Bower

A Senior Thesis submitted in partial fulfillment

of the requirements for graduation

in the Honors Program

Liberty University

Spring 2019

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Acceptance of Senior Honors Thesis

This Senior Honors Thesis is accepted in partial

fulfillment of the requirements for graduation from the

Honors Program of Liberty University.

______________________________

Scott Long, Ph.D.

Thesis Chair

______________________________

Daniel Joseph, Ph.D.

Committee Member

______________________________

Melesa Poole, Ph.D.

Committee Member

______________________________

Mark Ray Schmidt, Ph.D.

Assistant Honors Director

______________________________

Date

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Abstract

The origin of cryptography, the study of encoding and decoding messages, dates back to

ancient times around 1900 BC. The ancient Egyptians enlisted the use of basic encryption

techniques to conceal personal information. Eventually, the realm of cryptography grew

to include the concealment of more important information, and cryptography quickly

became the backbone of cyber security. Many companies today use encryption to protect

online data, and the government even uses encryption to conceal confidential

information. Mathematics played a huge role in advancing the methods of cryptography.

By looking at the math behind the most basic methods to the newest methods of

cryptography, one can learn how cryptography has advanced and will continue to

advance.

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Cryptography: Mathematical Advancements on Cyber Security

How does the government keep its secrets from prying eyes? How do companies

who promise to keep their customers’ personal information safe actually protect that

information? In most cases, the answer is cryptography, the study of encoding and

decoding information. Starting with rudimentary beginnings, cryptography evolved to

include complex mathematical techniques. As mathematicians discovered new aspects of

prime numbers and functions with unique properties, the government updated the

standard practice for encoding and decoding information in order to keep up with the

growing field of mathematics. Still today, researchers seek for new methods to add to

cryptography, and as mathematics continues to advance so will the area of cyber security.

History of Cryptography

Cryptography, the study of encryption and decryption, stems from the basic desire

for privacy. The most basic examples of the encryption and decryption process, though

rudimentary compared to modern methods, involve the concept of sending information to

a location without interception from an unintended source. The first known instance of

cryptography occurred around 1900 BC in ancient Egypt. The tomb of Khnumhotep II,

an Egyptian nobleman, contains several hieroglyphic symbols. However, the last sections

of the hieroglyphics include abnormal symbols replacing the usual symbols (Dooley,

2018). The unusual symbols obscure the meaning of the hieroglyphics by simply

switching commonly used symbols with unknown symbols. Though simple in nature, the

act of switching symbols to conceal a message is a basic form of encryption. Nonetheless,

one cannot determine with absolute certainty why Khnumhotep II hides the true meaning

of the inscriptions, so the reasoning behind the ambiguity remains unclear.

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Assyrian merchants during 1500 BC introduced the idea of using intaglios for

business transactions. An intaglio, an engraved figure in a stone or other hard material,

allowed the merchants to create a unique signature, which ensures that trading occurs

with the intended merchant (Dooley, 2018). The idea of using a unique signature foretells

the idea of digital signatures, a more mathematically complex way to ensure the

authenticity of a transaction.

Meanwhile, during the 7th century BC, a Greek poet named Archilochus

introduced the skytale, a cryptographic device consisting of a cylinder with a leather strip

(Dooley, 2018). Suppose the skytale allows for four letters written around the cylinder

and six letters written along the side, then skytale encrypts messages as follows:

Step 1. Create the message: “Send help as soon as possible”

Step 2. Write message along the side of the cylinder like the example below.

Step 3. Unwrap the leather strips to obtain the new encrypted message: “

slosepnsnaaidssbhspleooe ”

Notice: In order to decrypt the message, one must simply rewrap the leather strip

around the cylinder and read along the side. This method obviously presents several

limitations regarding the length of the original message and the ease of deciphering the

encrypted message but proves effective for the time period of the skytale’s use.

One of the most famous examples of early cryptography, the Caesar cipher,

emerged around 100 BC. Julius Caesar developed a shift cipher known as the Caesar

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cipher in order to protect confidential military information (Dooley, 2018). The cipher

takes the letters used in the message and shifts each letter of over three to the left, or 23 to

the right, according to the order of the alphabet. For example, if one wants to encrypt the

message “WE LEAVE AT SUNDOWN”, then by using the Caesar, he obtains the

following:

Original Message: WE LEAVE AT SUNDOWN

Encrypted Message: TB IBXSB XQ PRKALTK

Innovations to Cryptography

The ancient examples of cryptography certainly paved the way for more advanced

cryptographical techniques; however, several basic developments still needed

improvement. Even though the encryption techniques introduced thus far concealed

information, one could easily decrypt the message without much effort. In order to

further protect hidden messages, cryptographic techniques needed more complexity.

Leon Battista Alberti, an Italian Renaissance man, created the first polyalphabetic

cipher, known as the Alberti cipher, around 1467. The Alberti cipher involves two

concentric disks, one larger and one smaller, each divided into 24 sections. The stationary

larger disk contains the uppercase letters of the Latin alphabet, which consists of the

English alphabet minus J, U and W. In addition, Alberti also took out the letters H, K and

Y as he personally regarded them unnecessary, and he added the numbers 1 through 4 to

the outer disk, as well. The inner disk consists of the lowercase letters of the Latin

alphabet, which also disregards j, u and w from the English alphabet, and also contains

the symbol et, which most likely carries the meaning of the ‘&’ symbol (Dooley, 2018).

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In order to encode a message, one must simply rotate the inner disk and replace the letter

or number on the outer disk with the letter directly underneath on the inner disk.

As encryption and decryption techniques advanced, these methods began to

spread, especially with the invention of the printing press. Johannes Trithemius published

posthumously the first printed work on cryptography in 1518. His work entitled

Polygraphia is a composition of five books (Dooley, 2018). The books contain ancient

alphabets, invented alphabets, and examples of encoding messages. The printing of

cryptographical works allowed encryption and decryption techniques to spread farther

than ever before.

As the interest in creating cryptosystems grew, the interest in defeating

cryptosystems also emerged. Thus, the birth of cryptanalysis, the science of breaking

cryptographic algorithms, occurred during the 19th century. Edgar Allen Poe, the famous

American poet, showcased his skills in a Pennsylvania paper by solving submitted

ciphers. Poe’s efforts to break encryptions spread the interest of cracking encryption

techniques for entertainment. However, the increased interest in cryptanalysis pushed the

need for stronger encryption techniques (Dooley, 2018).

Eventually, governments also began to pay attention to cryptanalysis. During

World War I, the cryptanalysis section of the British Naval Intelligence decrypted about

15,000 German messages. This allowed the British Navy to play an important role in

defeating Germany during WWI. The realm of cryptography, now not only involved

methods for hiding important messages, but also involved the practice of breaking

encrypted messages. The ability to decrypt encoded messages allows one to gain an

advantage over enemies without their knowledge (Budiansky, 2016).

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Modern Advancements

As the interest in cryptography grew, the need for more complex methods also

arose. Thus, cryptographers sought to incorporate more difficult mathematical tactics in

order to protect their encryption techniques from adversaries (Middleton, 2017).

Whitfield Diffie and Martin Hellman, two prominent cryptographers of the 20th century,

incorporated the use of prime numbers into their encryption method, utilizing the

difficulty of factoring large prime numbers (Mollin, 2003). Diffie and Hellman created a

method to exchange keys, vital information used in encoding and decoding, called the

Diffie-Hellman key exchange. Essentially, the Diffie-Hellman key exchange involves the

use of a public key, which is a prime number known by the general public, a nonzero

integer agreed upon by the two parties exchanging information, and a unique secret key

for each of the two parties. Surprisingly, when using the Diffie-Hellman key exchange,

each party does not need to know the secret key of the other party in order perform the

required calculations. The specifications of the key exchange will be discussed later in

detail.

The 20th century also introduced the first commercially established cipher, known

by the name Lucifer. With the formation of the National Security Agency (NSA) during

1952, the American government began to search for more ways to ensure the protection

of domestic information (Yan, 2008). Several workers of the IBM, or International

Business Machines Corporation, created Lucifer and submitted the cipher to the NSA as a

possibility for the Data Encryption Standard (DES). After acceptance from the NSA,

Lucifer served as the basis for the DES, which became the government standard for

encryption (Yan, 2008).

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Although the discovery of a special curve, known as the elliptic curve, dates back

to the 18th century, the widespread implementation of elliptic curves into cryptography

did not occur until the beginning of the 21st century. The unique properties of addition

over an elliptic curve add important applications to cryptography. Elliptic curve

cryptography also enables more security with a smaller key size than the Diffie-Hellman

key exchange (Washington, 2003). The process of elliptic curve addition seems simple to

compute, however given large enough prime numbers the process becomes increasingly

more difficult. The NSA even accepted the use of elliptic curve cryptography as a

national standard for the encryption of information. However, the NSA announced the

plan to replace the use of elliptic curve cryptography with a newer cipher in 2015 due to

the impending threat of quantum computing (National Security Agency, 2015).

The Future of Cryptography

The unpredictability of the quantum realm makes the idea of quantum

cryptography a virtually unbreachable option for encoding important information. The

beauty of encoded data within a quantum state is that the moment an outside force tries to

intercept or read the encoded data, the data itself changes. Thus, eavesdropping is easily

detected the moment of occurrence. Quantum cryptography involves the use of photons

and a photon detector. Once the photon contains the hidden message, the photon detector

uses the random rotations of the polarization of the photon to transfer the message into

bits (Horodecki, 2010).

Since quantum cryptography is practically safe from any interference, it seems

completely infallible; yet, several complications make it hard to implement. One problem

with quantum cryptography is the short distance limitation for sending the data. An

CRYPTOGRAPHY 10

important aspect of quantum cryptography is the entanglement of photons, the idea that

two photons are linked even though there is no physical attachment between the two

photons (Horodecki, 2010). However, entanglement can only occur over a relatively short

distance. In 2017, scientists in China created a new record for the longest distance for

successfully transmitting entangled photons, 1203 kilometers, or approximately 760

miles (Chen, 2017). This distance is an incredible breakthrough, but room for major

improvement still exists.

Another area of concern with quantum cryptography is cost. For practical and

commercial uses of quantum cryptography, a reasonable cost and availability of the

required resources is nonexistent. Since much of quantum cryptography is still in the

developmental stages, many of the resources available to scientists and researchers are

not available generally. The network and equipment required for the quantum key

distribution, as well as the photon detector costs more than a business would typically

want to pay to add more security to the company (Sergienko, 2006). Once the

accessibility and the cost of the technology associated with quantum cryptography reach

a more affordable level, then wide spread implementation of quantum cryptography is

inevitable.

Private and Public Key Cryptography

In order to understand the role of mathematics in cryptography as a whole, one

should first attempt to understand how mathematics is used in the individual methods. An

important distinction to realize is the difference between private and public key

cryptography, but first one must consider the nature of a key itself and its purpose.

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A key is a piece of information that enables the sender of a message to encrypt the

message and the receiver of a message to decrypt the message (Hoffstein, Pipher, &

Silverman, 2008). For example, consider the Caesar cipher mentioned previously. This

particular cipher shifts each letter to the right by 23. Thus, the encryption key is 23.

However, the decryption key is less obvious than the encryption key in this example.

Remember in the previous discussion that shifting each letter of the alphabet to the right

by 23 is equivalent to shifting each letter over to the left by 3 taking in to consideration

the fact that there are 26 letters in the English alphabet. In order to return to the original

letters one can simply shift each letter in the encrypted message to the right 3 more times.

Therefore, the decryption key is 3. In most situations, the receiver needs to know only the

decryption key.

Private key encryption involves the use of one key. This key is both the

encryption and decryption key. Hence, the sender and receiver both need knowledge of

the same key. Private key encryption is a relatively simple process as only one key is

required for both encryption and decryption (Hoffstein, Pipher, & Silverman, 2008). In a

perfect world, private key cryptography is sufficient to protect important information.

However, most senders must worry about possible interference of their message. If the

secret key landed in the hands of an adversary, then he or she could steal or alter

confidential information with little effort. Even though private key encryption is simple

in theory, in actuality it brings many security threats that make the method not secure.

Public key encryption, a much safer alternative to private key encryption,

incorporates the use of two keys: one public key and one private key. The public key

functions as the encryption key which means that any individual can use the public key to

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encrypt a message, but only the owner of the private key, or the decryption key, can

decrypt the encoded messages (Wang, Xu, & Wang, 2016). Although public key

cryptography is more secure than private key cryptography, the method still involves

several weaknesses, so the process of storing the secret key must be secure.

As in the case of public key cryptography, if an adversary discovers the private

key, then he or she can also decrypt any encoded messages. As a result, many seek to

improve the means in which they exchange the secret key using prime numbers, elliptic

curves, and other more complex methods (Katz & Lindell, 2015). Another issue with

public key cryptography involves the public key, itself. The public key does not need to

be a secret key and will be published publicly. A third party, however, can intercept the

releasing of the public key and alter it. This third party can also create a unique private

key and decrypt any messages. In order to not raise any suspicions, the messages must

also continually be intercepted, decrypted, and then encrypted once again using the fake

public key.

Although this interception process certainly is possible, the more difficult the

encryption/decryption process, the harder for an adversary to pull off such a trick. Some

common examples of public key cryptography are the Diffie-Hellman key exchange and

the ElGamal key generator. Both encryption systems involve the use of prime numbers

and use the difficulty of factoring large primes to ensure more security (Katz & Lindell,

2015). Both processes will be discussed in detail later; however, one must first

understand some fundamental mathematical concepts.

Finite Fields

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Let 𝑆 be a set and 𝑆 × 𝑆 be the set of ordered pairs (𝑠, 𝑡) such that 𝑠, 𝑡 are

elements of 𝑆. A binary operation, ∗ , maps 𝑆 × 𝑆 → 𝑆. Note that the image of (𝑠, 𝑡) in

𝑆 × 𝑆 must also be an element in 𝑆. A group is a set 𝐺 such that the following properties

hold:

1. Associativity: For any 𝑎, 𝑏, 𝑐 𝜖 𝐺, 𝑎 ∗ (𝑏 ∗ 𝑐) = (𝑎 ∗ 𝑏) ∗ 𝑐.

2. Identity: For any 𝑎 𝜖 𝐺, there exists an element e such that 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎

3. Inverse: For any 𝑎 𝜖 𝐺, there exists an inverse element 𝑎−1 such that 𝑎 + 𝑎−1 =

𝑎−1 + 𝑎 = 𝑒.

The group is called an abelian group if the following property also holds:

4. Commutativity: For any 𝑎, 𝑏 𝜖 𝐺 , a ⁎ b = b ⁎ a.

A ring is a set 𝑅 with two binary operations + and ∙ that satisfy the following:

1. 𝑅 is an abelian group with respect to +.

2. The binary operation ∙ is associative.

3. The distributive property holds. Hence, for any 𝑎, 𝑏, 𝑐 𝜖 𝑅, 𝑎 ∙ ( 𝑏 + 𝑐 ) = 𝑎 ∙

𝑏 + 𝑎 ∙ 𝑐.

A ring is called a field if in addition the following also hold:

4. The binary operation ∙ is commutative.

5. The non-zero elements of 𝑅 form a group under ∙ .

A finite field is simply a field that contains finitely many elements. The finite field 𝔽𝑝 is

the finite field containing all the elements mod 𝑝 (Mullen & Panario, 2013; Cohen, Frey,

Avanzi, 2006).

Key Exchanges over Finite Fields

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The finite field 𝔽𝑝 provides many important applications to the realm of

cryptography. Perhaps one of the most relevant is the discrete logarithm problem. Note

that a primitive root is an element 𝑔 in 𝔽𝑝where the powers of 𝑔 generate the entire group

of 𝔽𝑝. The discrete logarithm problem is the difficulty of finding some 𝑥𝜖𝔽𝑝 such that

𝑔𝑥 = ℎ (𝑚𝑜𝑑 𝑝), where 𝑔 is a primitive root and h is any non-zero integer in 𝔽𝑝. The

discrete logarithm can be written as 𝑥 = log𝑔 ℎ. Note that ℎ = 𝑔 ∙ 𝑔 ∙ … ∙ 𝑔 (𝑚𝑜𝑑 𝑝)

for 𝑥 multiplications of 𝑔. Essentially, in order to find log𝑔 ℎ one must find how many

times g must be multiplied by itself in order to get ℎ (Hoffstein, Pipher, & Silverman,

2008).

An obvious way to compute the discrete logarithm is by testing out powers of 𝑔

one by one until some power 𝑖 such that 𝑔𝑖 = ℎ (𝑚𝑜𝑑 𝑝). For example, to find an 𝑎 such

that 2𝑥 = 7(𝑚𝑜𝑑 11), test the powers of 2, 20, 21, . . . 27, and eventually find that 27 =

7 (𝑚𝑜𝑑 11). However, this method becomes extremely difficult for any large prime

number. Both the Diffie-Hellman key exchange and the ElGamal key generator base their

computations on the difficulties surrounding the discrete logarithm (Katz & Lindell,

2015).

The Diffie-Hellman Key Exchange

Suppose two people, Alice and Bob, want to share a secret key, but Eve, an

outside adversary, can intercept any exchange between the two. By using the difficulty of

the discrete log problem to their advantage and the steps of the Diffie-Hellman key

exchange, Alice and Bob can avoid Eve’s obtaining of their key.

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Alice and Bob must first agree on some large prime p and a nonzero integer, g in

𝔽𝑝 which is available to the public, even to Eve. However, Alice also secretly chooses an

integer 𝑎 which she uses to calculate 𝐴 = 𝑔𝑎(𝑚𝑜𝑑 𝑝). Meanwhile, Bob also chooses a

secret integer b and calculates 𝐵 = 𝑔𝑏(𝑚𝑜𝑑 𝑝). Alice then sends 𝐴 to Bob, while Bob

sends 𝐵 to Alice. Alice takes 𝐵 and computes 𝐵𝑎, and Bob takes A and computes 𝐴𝑏.

Since 𝐵𝑎 = 𝐴𝑏 = 𝑔𝑎𝑏(𝑚𝑜𝑑 𝑝), both Alice and Bob receive the key, 𝑔𝑎𝑏(𝑚𝑜𝑑 𝑝)

without Eve also receiving the key (Katz & Lindell, 2015).

For example, assume Alice and Bob pick the prime number 𝑝 = 167 and 𝑔 = 2.

So, Alice takes 𝑝 and 𝑔 and computes 𝐴, using her secret number 𝑎 = 23 which only she

knows. Bob also takes 𝑝 and 𝑔 and computes 𝐵, using his secret number 𝑏 = 55 which

only Bob knows. So, 𝐴 = 223 ≡ 31(𝑚𝑜𝑑 167) and 𝐵 = 255 ≡ 50(𝑚𝑜𝑑 167). Alice

then sends Bob 𝐴 = 31 and Bob sends Alice 𝐵 = 50. With this new information, Alice

computes 𝐵𝑎 and Bob computes 𝐴𝑏 such that 𝐵𝑎 = 5021 ≡ 150 (𝑚𝑜𝑑 167) and 𝐴𝑏 =

3155 ≡ 150 (𝑚𝑜𝑑 167). Hence, both Alice and Bob obtain the key, 𝑘 = 150, without

anyone else also receiving the key. Note that an adversary, Eve, can obtain the values of

𝐴 and 𝐵, but neither Eve nor Bob knows the value of 𝑎, and neither Eve nor Alice knows

the value of 𝑏. This mean Eve would have to solve 2𝑎 ≡ 31 (𝑚𝑜𝑑 167) and 2𝑏 ≡

50 (𝑚𝑜𝑑 167) in order to find the values of 𝑎 and 𝑏, which becomes extremely difficult,

especially in a real life situation where the numbers are significantly larger.

The ElGamal Key Generator

ElGamal public key encryption closely resembles the Diffie-Hellman public key

exchange and also involves the discrete logarithm problem. The differences between the

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Diffie-Hellman and the ElGamal approach stem mostly from the calculation involved

with producing the shared key.

Say Alice chooses a prime number 𝑝 and a primitive root modulo 𝑝, 𝑔. She then

computes her public key by raising g to her private key, 𝑎. So, 𝐴 = 𝑔𝑎(𝑚𝑜𝑑 𝑝). Alice

then publishes this information so that anyone can encrypt a message using her public

key, but only Alice can decrypt the message using her private key. If Bob wants to send a

message 𝑚 to Alice, he must choose a random key 𝑘, which is called the ephemeral key,

and compute 𝐵1 = 𝑔𝑘(𝑚𝑜𝑑 𝑝) and 𝐵2 = 𝑚 ∙ 𝐴𝑘 = 𝑚 ∙ 𝑔𝑎𝑘(𝑚𝑜𝑑 𝑝). Bob sends Alice

both 𝐵1 and 𝐵2as a pair (𝐵1, 𝐵2) (Hoffstein, Pipher, & Silverman, 2008).

In order for Alice to decrypt the message, she must use her private key, a. Alice

begins by calculating 𝑥 = 𝐵1𝑎 = 𝑔𝑎𝑘(𝑚𝑜𝑑 𝑝). Then, she finds 𝑥−1 = 𝑏 (𝑚𝑜𝑑 𝑝) and

computes 𝐵2 ∙ 𝑥−1 = 𝑚 ∙ 𝑥 ∙ 𝑥−1 = 𝑚. Hence, Alice decrypts the encrypted message to

obtain the original message 𝑚 (Hoffstein. Pipher, & Silverman, 2008).

As an example, let Alice choose 𝑝 = 179 and 𝑔 = 2. She then chooses her

private key, 𝑎 = 63 and computes 𝐴 = 263 ≡ 63(𝑚𝑜𝑑 179). She then releases 𝑝, 𝑔, and

𝐴 to the public. Now, suppose Bob wants to send Alice a message 𝑚 = 123, so he picks

a random integer 𝑘 = 131 and computes 𝐵1 = 2131 ≡ 35(𝑚𝑜𝑑 179) and 𝐵2 = 123 ∙

62131 = 74(𝑚𝑜𝑑 179). Bob then sends Alice the pair (35,74). Once Alice receives the

pair, she calculates 𝑥 = 3563 ≡ 69(𝑚𝑜𝑑 179) and 𝑥−1 = 96(𝑚𝑜𝑑 179). By using 𝐵2

Alice can decrypt the encrypted message to find the original message 𝑚. Alice computes

𝐵2 ∙ 𝑥−1 = 74 ∙ 96 ≡ 123(𝑚𝑜𝑑 179). Hence, Alice successfully decrypts the message,

𝑚 = 123, sent to her by Bob.

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Consider the issues with public key cryptography discussed earlier. Both the

Diffie-Hellman key exchange and the ElGamal key generator help combat these issues;

however, the issues do not disappear completely. In the case of the Diffie-Hellman key

exchange, if Eve wants to obtain the private key shared by Alice and Bob, she needs both

Alice’s secret integer 𝑎 and Bob’s secret integer 𝑏. Notice the difficulty in Eve obtaining

both 𝑎 and 𝑏 since neither Alice nor Bob know both 𝑎 and 𝑏 (Katz & Lindell, 2015).

Also, in the example for Diffie-Hellman, the prime number chosen is 167, but in real life

examples, the prime number chosen are extremely large making the computations for Eve

even more difficult. Nonetheless, large primes also make the computations difficult for

Alice and Bob as well.

Regarding the ElGamal key generator, for Eve to decipher a message m, she not

only deals with the repercussions of the discrete logarithm problem but also with the

difficulty of modular inverses. Alice may still have a difficult time computing the

modular inverse of 𝑥 even though she also has her secret key, 𝑎. Eve, without both Alice

and Bob’s secret keys, faces the daunting task of finding the correct exponent of 𝑔 in

addition to then calculating the modular inverse of an exceedingly large number

(Hoffstein. Pipher, & Silverman, 2008). Notice also that the message 𝑚 is a numerical

value, so for longer messages the difficulty of the computations increases.

Elliptic Curves

The use of elliptic curves in cryptography greatly improved the security of

encryption ciphers, but what exactly is an elliptic curve? An elliptic curve over 𝔽𝑝 is the

set of solutions to an equation of the form 𝑥3 + 𝑎𝑥 + 𝑏 where 𝑎, 𝑏 are elements of 𝔽𝑝 and

4𝑎3 + 27𝑏2 ≠ 0. The form 𝑥3 + 𝑎𝑥 + 𝑏 of an elliptic curve is known as the Weierstrass

CRYPTOGRAPHY 18

form named after the German mathematician who discovered the form, Karl Weierstrass.

The condition 4𝑎3 + 27𝑏2 ≠ 0 is also known as the discriminant of an elliptic curve

(Silverman, 2009).

An important attribute of elliptic curves is addition over the curve. The binary

operation ⊕ is called elliptic curve addition. Graphically, given two points 𝑃 and 𝑄 on

an elliptic curve 𝐸, we obtain 𝑃 ⊕𝑄 by connecting a line 𝐿 from 𝑃 to 𝑄 and finding the

third intersection point labeled as 𝑅. By reflecting 𝑅 across the 𝑥-axis, one obtains −𝑅

for which 𝑃 ⊕𝑄 = −𝑅. Also, call 𝑂 the point at infinity, the identity for 𝐸, and say it

exists at every vertical line (Hoffstein, Pipher, & Silverman, 2008).

Given an elliptic curve 𝐸 over 𝔽𝑝 the following properties are also true:

1. Identity: 𝑃⊕𝑂 = 𝑂⊕ 𝑃 = 𝑃 for all 𝑃 in 𝐸(𝔽𝑝).

2. Inverse: 𝑃 ⊕ (−𝑃) = 𝑂 for all 𝑃 in 𝐸(𝔽𝑝).

3. Associativity: (𝑃 ⊕ 𝑄)⊕𝑅 = 𝑃⊕ (𝑄 ⊕ 𝑅) for all 𝑃, 𝑄, 𝑅 in 𝐸(𝔽𝑝).

4. Commutativity: 𝑃 ⊕𝑄 = 𝑄 ⊕𝑃 for all 𝑃, 𝑄 in 𝐸(𝔽𝑝).

Based on the previous definition of an abelian group, one can see that 𝐸 is an

abelian group under elliptic curve addition. However, computing ⊕ graphically becomes

𝑃 𝑄

−𝑅

𝑅 = 𝑃⊕𝑄

Elliptic Curve Addition

CRYPTOGRAPHY 19

difficult and impractical when dealing with many additions. Thus, one can also use an

elliptic curve addition algorithm (Hoffstein, Pipher, & Silverman, 2008).

Given an elliptic curve 𝐸: 𝑦2 = 𝑥3 + 𝑎𝑥 + 𝑏 over 𝔽𝑝 and two points 𝑃1, 𝑃2 ϵ

𝐸(𝔽𝑝),

1. If 𝑃1 = 𝑂, then 𝑃1⊕𝑃2 = 𝑃2.

2. If 𝑃2 = 𝑂, then 𝑃1⊕𝑃2 = 𝑃1.

3. Else, let 𝑃1 = (𝑥1, 𝑦1) and 𝑃2 = (𝑥2, 𝑦2).

4. If 𝑥1 = 𝑥2and 𝑦1 = −𝑦2, then 𝑃1⊕𝑃2 = 𝑂.

5. Else, define λ as

𝜆 =

{

𝑦2 − 𝑦1𝑥2 − 𝑥1

𝑖𝑓 𝑃1 ≠ 𝑃2

3𝑥1+𝑎2

2𝑦_1 𝑖𝑓 𝑃1 = 𝑃2

where 𝑥3 = 𝜆2 − 𝑥1 − 𝑥2 and 𝑦3 = 𝜆(𝑥1 − 𝑥3) − 𝑦1. Thus, 𝑃1⊕𝑃2 = (𝑥3, 𝑦3). For

example: Let 𝐸(𝔽23): 𝑦2 = 𝑥3 + 𝑥 + 1, 𝑃1 = (1,16), and 𝑃2 = (11,20). Since 𝑃1 ≠ 𝑃2,

𝜆 =20−16≡4

11−1≡10 = 4 ∙ 10−1 = 4 ∙ 7 ≡ 5(𝑚𝑜𝑑 23). Then use λ to compute 𝑥3 and 𝑦3

where 𝑥3 = 52– 1– 11 = 13(𝑚𝑜𝑑 23) and 𝑦3 = 5(1 − 13) − 16 = 16(𝑚𝑜𝑑 23). Thus,

𝑃1⊕𝑃2 = (13,16).

Recall that a Weierstrass equation of an elliptic curve is an equation of the form

𝑦2 = 𝑥3 + 𝑎𝑥 + 𝑏. However, this form is not defined for 𝔽2 or 𝔽3, so one must also

consider a more general form of the Weierstrass equation suitable for all 𝔽𝑝. The long

Weierstrass form of an elliptic curve over 𝔽𝑝 is

𝐸: 𝑦2 + 𝑎1𝑥𝑦 + 𝑎2 = 𝑥3 + 𝑎32 + 𝑎4𝑥 + 𝑎5 where 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5 ϵ 𝔽𝑝.

CRYPTOGRAPHY 20

The long Weiestrass form works over any 𝔽𝑝. Although, for the purposes of

cryptography, the short Weierstrass equation is sufficient (Blake, Seroussi, & Smart,

2005). In fact, there exists an isogeny between the long Weierstrass form and the short

Weierstrass form, meaning that the short Weierstrass from preserves the structure of an

long form of an elliptic curve and can be used for cryptography. Given E and E’ as two

elliptic curves over 𝔽𝑝, 𝜙: 𝐸 → 𝐸′ is an isogeny if 1) 𝑥 = 𝑢2𝑥′ + 𝑟 and 2) 𝑦 = 𝑢3𝑦′ +

𝑢2𝑠𝑥′ + 𝑡 where (𝑥, 𝑦) ∈ 𝐸, (𝑥′, 𝑦′) ∈ 𝐸′, and 𝑢, 𝑠, 𝑡, 𝑟 ∈ 𝔽𝑝 (Silverman, 2009).

Now, the fact that an elliptic curve with short Weierstrass form and an elliptic

curve with long Weierstrass form an isogeny can be proven in the following manner:

Proof. Let E’ be an elliptic curve with long Weiestrass form and E be an elliptic

curve with short Weierstrass form over 𝔽𝑝 such that,

𝐸′: (𝑦′)2 + 𝑎1𝑥′𝑦′ + 𝑎2′ = (𝑥′)3 + (𝑎3

′ )2 + 𝑎4𝑥′ + 𝑎5 and

𝐸: 𝑦2 = 𝑥3 + 𝑐1𝑥 + 𝑐2.

Beginning with E’, the first goal is to manipulate E’ in such a way that y can replace

𝑢3𝑦′ + 𝑢2𝑠𝑥′ + 𝑡 for some 𝑢, 𝑠, 𝑡 ∈ 𝔽𝑝.

(𝑦′)2 + 𝑎1𝑥′𝑦′ + 𝑎2𝑦′ = (𝑥′)3 + (𝑎3𝑥

′)2 + 𝑎4𝑥′ + 𝑎5

(𝑦′)2 + (𝑎1𝑥′ + 𝑎2)𝑦

′ +(𝑎1𝑥+𝑎2)

2

4= (𝑥′)3 + (𝑎3𝑥′)

2 + 𝑎4𝑥′ + 𝑎5 +(𝑎1𝑥+𝑎2)

2

4

(𝑦′ +𝑎1𝑥

′+𝑎2

2)2

= (𝑥′)3 + (𝑎3 +𝑎12

4) (𝑥′)2 + (𝑎4 +

2𝑎1𝑎2

4) 𝑥′ + (

𝑎22

4+ 𝑎5)

By setting 𝑦 = 𝑦′ + 𝑎1

2𝑥′ +

𝑎3

2, the equation becomes

𝑦2 = (𝑥′)3 + (4𝑎3+𝑎1

2

4) (𝑥′)2 + (

2𝑎4+𝑎1𝑎2

2) 𝑥′ +

4𝑎5+𝑎22

4.

CRYPTOGRAPHY 21

In order to simplify the equation for future calculations, set 4𝑎3 + 𝑎12 = 𝑏1, 2𝑎4 +

𝑎1𝑎2 = 𝑏2, and 4𝑎5 + 𝑎22 = 𝑏3. Thus, the equation is

𝑦2 = (𝑥′)3 +𝑏1

4(𝑥′)2 +

𝑏2

2𝑥′ +

𝑏3

4.

The next objective of the proof is to manipulate the equations further so that x can replace

𝑢2𝑥′ + 𝑟 for some 𝑢, 𝑟 ∈ 𝔽𝑝.

𝑦2 = ((𝑥′)3 +𝑏1

4(𝑥′)2 +

𝑏1

48𝑥′ +

𝑏1

1728) +

𝑏2

2𝑥′ +

𝑏3

4−

𝑏1

48𝑥′ −

𝑏1

1728

𝑦2 = (𝑥′ +𝑏1

12)3

− (𝑏12

48𝑥′ −

𝑏2

2𝑥′ +

𝑏13

576−𝑏1𝑏2

24) −

𝑏13

1728+𝑏3

4+

𝑏13

576−𝑏1𝑏2

24

𝑦2 = (𝑥′ +𝑏1

12)3

− (𝑏12−24𝑏2

48) (𝑥′ +

𝑏1

12) −

𝑏13+3𝑏1

3+432𝑏3−72𝑏1𝑏2

1728

𝑦2 = (𝑥′ +𝑏1

12)3

− (𝑏12−24𝑏2

48) (𝑥′ +

𝑏1

12) −

𝑏13−36𝑏1𝑏2+216𝑏3

864

By letting 𝑥 = 𝑥′ +𝑏1

12,

𝑦2 = 𝑥3 −𝑏12−24𝑏2

48𝑥 −

𝑏13−36𝑏1𝑏2+216𝑏3

864 .

Notice for 𝑐1 =24𝑏2−𝑏1

2

48 and 𝑐2 =

𝑏13−36𝑏1𝑏2+216𝑏3

864,

𝑦2 = 𝑥3 + 𝑐1𝑥 + 𝑐2 = 𝐸.

The substitutions 𝑦 = 𝑦′ + 𝑎1

2𝑥′ +

𝑎3

2 and 𝑥 = 𝑥′ +

𝑏1

12 follow the definition of an

isogeny where 𝑢 = 1, 𝑡 =𝑎3

2, 𝑠 =

𝑎1

2, and 𝑟 =

𝑏1

12. Thus, there exists an isogeny such that

𝜙: 𝐸 → 𝐸′ and (𝑥, 𝑦) ↦ (𝑥′ +𝑏1

12, 𝑦′ +

𝑎1

2𝑥′ +

𝑎3

2). Therefore, the structure of an elliptic

curve is preserved when using the short Weierstrass form, and this form can be used for

elliptic curve cryptography (Silverman, 2009).

CRYPTOGRAPHY 22

An important attribute of elliptic curves is the fact that the curve is non-singular

and smooth, which ensures that the curve contains unique solutions. Without the

characteristic of being non-singular, elliptic curves could not be used for cryptography.

Hence, the condition 4𝑎3 + 27𝑏2 ≠ 0, or the discriminant, assures that the curve is, in

fact, non-singular (Cohen, Frey, & Avanzi, 2006). Essentially, the discriminant of an

elliptic curve implies that there is a tangent line at every point, thus E is non-singular and

smooth. One can prove that the discriminant is 4𝑎3 + 27𝑏2 ≠ 0 in the following

manner:

Proof. Let 𝑓(𝑥, 𝑦) = 𝑦2 − 𝑥3– 𝑎𝑥 − 𝑏. Then,𝜕𝑓

𝜕𝑥 = −3𝑥2 − 𝑎 and

𝜕𝑓

𝜕𝑦= 2𝑦.

Hence, 𝜕𝑦

𝜕𝑥=

3𝑥^2 + 𝑎

2𝑦, and notice that ∇𝑓 = ⟨3𝑥2 + 𝑎, 2𝑦⟩. 𝐸 is smooth if ∇𝑓 exists and is

nonzero at ⟨𝜕𝑓

𝜕𝑥,𝜕𝑓

𝜕𝑦⟩. So, find where

𝜕𝑓

𝜕𝑥 and

𝜕𝑓

𝜕𝑦 are both equal to zero. When

𝜕𝑓

𝜕𝑥= 0, 𝑎 =

−3𝑥2 and when 𝜕𝑓

𝜕𝑦= 0, 𝑦 = 0. Thus, when 𝑎 = −3𝑥2 and 𝑦 = 0,

𝑓(𝑥, 0) = −𝑥3 − (−3𝑥2)𝑥 − 𝑏 = 2𝑥3– 𝑏.

In order to ensure that 𝐸 is non-singular, set 𝑓 = 0 and then restrict the result. Hence,

2𝑥3 − 𝑏 = 0 and thus 𝑥3 =𝑏

2. Notice that 𝑎 = 3𝑥2, so 𝑎3 = −27𝑥6 and 𝑥6 = −

𝑎3

27.

Therefore,

(𝑏

2)2

= 𝑥6 = −𝑎3

27 .

So, – 27𝑏2 = 4𝑎3 and thus 4𝑎3 + 27𝑏2 = 0.

Remember that in order for 𝐸 to be non-singular, one must restrict the values 𝑎 and 𝑏

such that 4𝑎3 + 27𝑏2 ≠ 0. Therefore, since the restriction ensures that ∇𝑓 exists and is

nonzero at every point of the curve, the discriminant of an elliptic curve implies that there

CRYPTOGRAPHY 23

is a tangent line at every point; thus, E is non-singular and smooth. Now, one can

properly understand the cryptographic applications of elliptic curves.

A similar discrete logarithm problem occurs with elliptic curves. However, the

discrete logarithm problem for elliptic curves involves elliptic curve addition instead of

multiplication. Thus, the elliptic curve discrete logarithm problem is the problem of

finding some 𝑛 such that 𝑄 = 𝑃 ⊕ 𝑃⊕. . .⊕ 𝑃 where 𝑃 is added to itself 𝑛 times

(Cohen, Frey, & Avanzi, 2006). Formally, Given an elliptic curve over 𝔽𝑝 and points 𝑃,

𝑄 in 𝔽𝑝, the elliptic curve discrete logarithm problem is the problem of finding an 𝑛 such

that 𝑄 = 𝑛𝑃, where one writes 𝑛 = log𝑃 𝑄. Next, consider how the elliptic curve discrete

logarithm problem changes the approach to the Diffie-Hellman key exchange and the

ElGamal key generator.

Diffie-Hellman Elliptic Curve Key Exchange

Elliptic curves can also be used to exchange keys using the Diffie-Hellman key

exchange. However, instead of choosing a prime number 𝑝 and a nonzero integer 𝑔 in

𝔽𝑝, Alice and Bob must choose 𝑝, an elliptic curve over a finite field 𝐸(𝔽𝑝), and a point

𝑃 in 𝐸(𝔽𝑝) (Cohen, Frey, & Avanzi, 2006). Note that the Diffie-Hellman key exchange

with elliptic curves uses elliptic curve addition, not multiplication as in the key exchange

for 𝔽𝑝.

After Alice and Bob agree on 𝐸(𝔽𝑝) and 𝑃, Alice picks a secret integer 𝑎 and she

computes 𝑄𝑎 = 𝑎𝑃. Meanwhile, Bob picks a secret integer 𝑏 and computes 𝑄𝑏 = 𝑏𝑃.

Alice then sends 𝑄𝑎 to Bob, and Bob sends 𝑄𝑏 to Alice. Both Alice and Bob use their

CRYPTOGRAPHY 24

respective secret integers 𝑎 and 𝑏 to compute 𝑎𝑄𝑏 = 𝑎𝑏𝑃 = 𝑏𝑄𝑎, which becomes a

secret key shared by Alice and Bob (Cohen, Frey, & Avanzi, 2006).

Specifically, say Alice and Bob agree on 𝐸(𝔽3023): 𝑦2 = 𝑥3 + 𝑥 + 2547 and 𝑃 =

(2237,2480) where 𝑃𝜖𝐸(𝔽3023). Alice then chooses a secret integer 𝑎 = 2313 and

calculates 𝑄𝑎 = 2313𝑃 = (934, 29). Meanwhile, Bob also chooses a secret integer 𝑏 =

1236 and calculates 𝑄𝑏 = 1236𝑃 = (1713,1709). Then, Alice sends 𝑄𝑎 to Bob, and

Bob sends 𝑄𝑏 to Alice. By using their respective secret integers, 𝑎 = 2313 and 𝑏 =

1236, both Alice and Bob obtain the secret key 2313𝑄𝑏 = 1236𝑄𝑎 = (2537,1632).

Even though both the Diffie-Hellman key exchange over 𝔽𝑝 and 𝐸(𝔽𝑝) involve

similar processes, the recommended key size for the elliptic Diffie-Hellman key

exchange is significantly smaller than that for the key exchange over 𝔽𝑝. According to

NIST, a 3072-bit key or larger is recommended for the Diffie-Hellman key exchange

over 𝔽𝑝, while only a 384-bit is recommended for the elliptic Diffie-Hellman key

exchange (Barker, 2016). Thus, the Diffie-Hellman elliptic curve key exchange can

provide sufficient security with a significantly smaller key than the Diffie-Hellman key

exchange over finite fields.

The ElGamal Elliptic Curve Key Generator

Elliptic curves also provide application to ElGamal cryptosystems. Alice chooses

a prime number 𝑝, an elliptic curve 𝐸(𝔽𝑝), and a point 𝑃 in 𝐸(𝔽𝑝). After she chooses a

secret key 𝑎, she then computes 𝑄𝑎 = 𝑎𝑃 and publishes 𝑄𝑎, along with 𝑝, 𝐸(𝔽𝑝), and 𝑃.

Now, if Bob wants to send a message 𝑀 in 𝐸(𝔽𝑝) to Alice, he chooses a random integer

CRYPTOGRAPHY 25

𝑘 to be his ephemeral key and then computes 𝐵1 = 𝑘𝑃 and 𝐵2 = 𝑀⊕ 𝑘𝑄𝑎. Bob sends

(𝐵1, 𝐵2) to Alice. Alice then takes the pair from Bob and using her secret key a computes

𝐵2 − 𝑎𝐵1 = (𝑀⊕ 𝑘𝑄𝑎) − 𝑎𝑘𝑃 = 𝑀⊕ 𝑎𝑘𝑃⊕−𝑎𝑘𝑃 = 𝑀.

Thus, Alice receives the original message 𝑀 (Hoffstein, Pipher, & Silverman, 2008).

Now, suppose Alice chose 𝑝 = 3023, 𝐸(𝔽3023): 𝑦2 = 𝑥^3 + 𝑥 + 2547, and

𝑃 = (2237,2430). Alice chooses a secret key 𝑎 = 2313, and computes 𝑄𝑎 = 2313𝑃 =

(934,29) which Alice makes available to the public along with 𝑝,𝐸(𝔽3023), and 𝑃. Bob

decides that he wants to send Alice a message 𝑀 = (2181,4000) in 𝐸(𝔽3023), so he

chooses a random integer 𝑘 = 1236 and calculates 𝐵1 = 1236𝑝 = (1713,1709) and

𝐵2 = (2181,400) ⊕ 1236𝑄𝑎 = (2181,400)⊕ (2537,1632) = (2720, 452). Bob

sends Alice (𝐵1, 𝐵2) where 𝐵1 = (1713, 1709) and 𝐵2 = (2720, 452). Alice receives

(𝐵1, 𝐵2) and uses her secret key a to obtain the original message 𝑀.

𝐵2 − 𝑎𝐵1 = (2720, 452) ⊕ −2313𝐵1 = (2720, 452)⊕ (2537,−16) = (2181, 400)

Since (2181, 400) = 𝑀, Alice obtains the original message.

Advantages and Disadvantages of Elliptic Curve Cryptography

After looking at the application of elliptic curves to the Diffie-Hellman key

exchange and the ElGamal key exchange, the efficiency of elliptic curves in securing

private information is apparent. Elliptic curves add a unique layer to cryptography and

elliptic curve addition provides a way to apply methods in 𝔽𝑝 to 𝐸(𝔽𝑝) instead (Xiong,

Qin, & Vasilakos, 2017). Though the general public may be unaware of elliptic curves

and their applications, their importance to cryptography is apparent just with their

addition to public key exchanges.

CRYPTOGRAPHY 26

However, even elliptic curve cryptography faces several disadvantages. First of

all, the computations with elliptic curve cryptography involve more complex

mathematics. While the computations surrounding public key cryptography over finite

fields make sense even without a full understanding of the mathematics behind the

system, understanding the process of elliptic curve cryptography and the addition process

comes less from common knowledge and more from extensive research (Silverman,

2009). Elliptic curve addition adds multiple steps to the encryption and decryption

process, so the likelihood of implementation errors increases. Also, elliptic curve addition

significantly increases the key size compared to basic multiplication making the

computations even more difficult.

Quantum Mechanics

Even with the mathematical advancements thus far, security breaches still exist as

none of the cryptographical methods are unbreakable. However, the quantum realm

offers a unique perspective on cryptography and multiple other areas of study (Sergienko,

2006). Before diving into the complexities of quantum cryptography, one should seek

understanding of the laws associated with the quantum realm.

Quantum mechanics describes the actions of atoms and subatomic particles.

When looking at the nature of the quantum realm, one must understand that particles act

according to a completely different set of rules than the visible world (Griffiths, 2006).

Consider a car driving down a road; one can easily find the position of the car at a certain

time, as well as the velocity and mass by taking basic measurements. Quantum particles,

however, do not act according to classic physics.

CRYPTOGRAPHY 27

German physicist, Werner Heisenberg, studied the nature of particles and

discovered a key component of quantum particles: the momentum and position of a

particle cannot be known with full certainty. The uncertainty principle states that the

inaccuracy of the position of a particle multiplied by the inaccuracy of the momentum of

a particle must be larger than ℎ

4𝜋, where ℎ is Planck’s constant (Griffiths, 2006).

Essentially, the uncertainty principle says the more one knows about the position of a

particle, the less he or she knows about the momentum and vice versa. Even the very act

measuring a quantum system disturbs the system, itself. Thus, one cannot know the

precision of the position and momentum of a particle with absolute certainty.

Mathematically, the uncertainty principle says 𝛥𝑥𝛥𝑝 >ℎ

4𝜋 where 𝑥 is the position and 𝑝

is the momentum of a particle (Griffiths, 2006).

Think of electrons circling a nucleus. Knowing that opposite charges attract, one

might think that the electrons would be attracted all the way to the positively charged

nucleus. However, the certainty of the electrons’ location close to the nucleus implies that

the uncertainty of the electrons’ momentum would be enormous. Thus, the electrons may

be moving so fast that they leave the atom entirely (Griffiths, 2006; Horodecki, 2010).

Remember the example of the car driving down a road. The observability of the position

and momentum of the car makes it hard to believe that the particles which make up the

car are acting in a manner contrary to the observable car.

Particles at the subatomic level move similar to waves. An important property of

waves is that waves can be added together to produce another wave. This property can

also be applied to quantum states. Two quantum states can be added together to produce

CRYPTOGRAPHY 28

another quantum state. This idea that particles can be in two states at the same time id

called superposition. One practical example of superposition in motion is the qubit. A

classical bit, or binary digit, can only appear as one of two states, usually either 1 or 0. In

contrast, qubits can also appear as the superposition of 1 and 0 thereby existing as two

states at the same time (Sergienko, 2006).

An interesting aspect of the quantum realm is the phenomenon known as

entanglement. Entanglement occurs when the quantum state of a pair or group of particles

exist dependently upon one another (Griffiths, 2006). Entangled particles essentially

cannot be described without each other, and a correlation exists between the particles

even over a physical distance.

To understand the idea of entangled particles, picture a pair of gloves, one right-

handed glove and one left-handed glove. Now, suppose that an individual places each

glove in a box and ships them to two different locations where Person 1 waits at one

location and Person 2 waits at the other location. Once both individuals receive their

boxes, Person 1 opens his boxes to reveal a left-handed glove, so Person 2’s box must

contain a right-handed glove. Notice that the moment that Person 1 opened his box and a

left-handed glove was revealed, Person 2 could only have a right-handed glove.

Entangled particles act in the same manner, once a certain physical attribute of one

particle is revealed, the other particle(s) act in accordance (Griffiths, 2006; Horodecki,

2010). The idea of entanglement and superposition plays a crucial role in quantum

cryptography.

Quantum Cryptography

CRYPTOGRAPHY 29

Several features of quantum mechanics provide useful applications to

cryptography. A well-known application of quantum mechanics to cryptography is with

quantum key distributions. The ability to send information through quantum particles

eliminates many of the issues with the usual key distribution methods (Horodecki, 2010).

Remember the very act of measuring particles disturbs the systems leading to even more

uncertainty about the position or momentum of a particle. Hence, the moment that an

adversary attempts to eavesdrop or intercept the key exchange between two parties, they

can immediately detect a disturbance.

Most quantum key distributions use the polarization of photons in order to

exchange information. Light emitting diodes, or LEDs, enable an individual to control the

creation of the photons by making one single photon at a time. Also, a polarization filter

allows for only photons with a certain polarization to pass through the filter while

blocking all other photons from coming through (Sergienko, 2006). The difficulty with

measuring polarization is that once the photons are polarized, they cannot be measured

again without disrupting the polarization of the photons, itself, in the process. Quantum

key exchanges rely on attaching information to a photon and knowing the original

polarization of the photon (Sergienko, 2006). How can one know the original

polarization of the photon when the very act of measuring the polarization disturbs the

polarization itself?

First, one must determine which types of polarizations to use. The two most well-

known polarization bases are rectilinear and diagonal polarization, and it is important to

note that a polarization base must consist of two orthogonal states. The rectilinear basis

includes a vertical polarization of 0°, meaning the photons move in an up and down

CRYPTOGRAPHY 30

oscillating motion, and a horizontal polarization of 90°, meaning the photons move in a

side to side oscillating motion. The diagonal basis uses a polarization of 45° and 135°.

Notice that these angles are also orthogonal, so a polarization of 45° correlates to a wave-

like motion at 45°, and a polarization of 135° correlates to a wake-like motion at 135°

(Sergienko, 2006).

Rectilinear Basis: Vertical vs. Horizontal Polarization

Diagonal Basis: 45° vs. 135° Polarization

The first quantum cryptography protocol, known as BB84, uses two orthogonal

states, usually the rectilinear basis and the diagonal basis. The protocol then assigns a 0

bit to both 0° and 45° and a 1 bit to both 90° and 135° (Sergienko, 2006). Also, the

rectilinear basis is labeled by the symbol + and the diagonal basis is labeled by the

CRYPTOGRAPHY 31

symbol ×.

To better understand the steps of the quantum key exchange, suppose Alice and

Bob want to share a message through some quantum communication channel, likely an

optical fibre or a vacuum, and any adversary, Eve, can try any approach to mess up the

communication within the quantum channel. Also, Alice decides to use the rectilinear

basis and the diagonal basis as her orthogonal bases. She starts by randomly choosing a 0

or 1 bit and then randomly choosing a basis as well. Say Alice randomly chooses a 1 bit

and the diagonal basis. She then transmits a photon to Bob in the horizontal state

according to the diagonal basis at 135° (Horodecki, 2010).

Bob receives the photon and randomly chooses either a rectilinear or diagonal

filter to measure the photon assuming that Bob is unaware of what basis Alice used to

polarize the photons. When Bob finishes measuring the photons sent by Alice, he

communicates to Alice. Alice and Bob then discuss over the channel where Bob says the

basis he used to measure each photon and Alice says whether Bob is correct or incorrect

in lining up with how she polarized the photons. They both discard the bits where Bob

used a different basis and thus, the key is the remaining bits (Horodecki, 2010).

Specifically, say Alice and Bob decide to use the quantum key exchange with the

rectilinear and diagonal bases. Alice sends eight photons randomly using the method

0 1

+

×

CRYPTOGRAPHY 32

described above, and Bob measures the photons he receives randomly. They receive the

following information after discussing through a quantum communication channel.

Alice’s Bit 0 0 1 0 1 1 1 0

Alice’s Random

Basis

+ × × + + + × +

Alice’s Angle 0° 45° 135° 0° 90° 90° 135° 0°

Bob’s Random

Measurement

+ + × × + × + +

Bob’s Angle 0° 90° 135° 45° 90° 135° 90° 0°

Matches X X X X

Shared Key 0 1 1 0

So, Alice and Bob both receive the shared secret key 0110. Also, notice that for the

photon polarizations where the basis does not line up, the photon detector randomly

chooses between vertical and horizontal polarization (Sergienko, 2006).

Conclusion

After looking through the different techniques of cryptography, one clearly sees

how mathematics has impacted the realm of cryptography. However, as mathematics

evolves and the level of security grows with each newly implemented cryptographical

method, adversaries also grow in knowledge and understanding of the underlying process

of security. Thus, just as mathematics is always evolving, the application of mathematics

to cyber security must also be growing. Each cryptographical method discussed, contains

advantages and disadvantages concerning its respective encoding and decoding

processes. Public key cryptography of a finite field proves the simplest to calculate and

understand but requires a larger key size in order to reach the level of security of elliptic

curve cryptography. Although elliptic curve cryptography allows for a smaller key size,

the complex mathematical techniques leave room for calculation errors and confusion.

CRYPTOGRAPHY 33

Quantum cryptography seems to excel in many areas where the other methods fail.

However, the distance over which quantum cryptography can take place is still lacking.

Considering that researchers have only just scratched the surface of the quantum realm,

as time goes on and knowledge progresses, cryptographical techniques will continue to

advance.

CRYPTOGRAPHY 34

References

Barker, E.(2016). Recommendation for key management part 1. NIST Special

Publication. doi:10.6028/nist.sp.800-57pt1r4

Blake, I. F., Seroussi, G., & Smart, N. P. (2005). Advances in elliptic curve

cryptography. New York: Cambridge University Press.

Budiansky, S. (2016). Code warriors: NSA's codebreakers and the secret intelligence

war against the Soviet Union. New York: Vintage Books.

Chen, N. (2017). China' s quantum satellite establishes photon entanglement over 1,200

km. Retrieved from http://english.cas.cn/newsroom/news/

201706/t20170619_178279.shtml.

Cohen, H., Frey, G., & Avanzi, R. (2006;2005;). Handbook of elliptic and hyperelliptic

curve cryptography (1st ed.). Boca Raton, FL: Chapman & Hall/CRC.

Dooley, J. F. (2018). History of cryptography and cryptanalysis: codes, ciphers, and

their algorithms. Cham: Springer Nature.

Griffiths, D. (2006). Introduction to quantum mechanics (Second edition.). Upper Saddle

River, NJ: Pearson Prentice Hall.

Hoffstein, J., Pipher, J., & Silverman, J. H. (2008). Introduction to mathematical

cryptography. New York: Springer-Verlag New York.

Horodecki, R. (2010). Quantum cryptography and computing: theory and

implementation. IOS Press.

Katz, J., & Lindell, Y. (2015). Introduction to modern cryptography (Second ed.).

London: CRC.

CRYPTOGRAPHY 35

Middleton, B. (2017). A history of cyber security attacks: 1980 to present (1st ed.).

Philadelphia: CRC.

Mollin, R. A. (2003). RSA and public-key cryptography (1st ed.). Boca Raton, FL:

Chapman & Hall/CRC.

Mullen, G., & Panario, D. (2013). Handbook of finite fields. Boca Raton, FL: CRC.

National Security Agency: central security service defending our nation. (2015).

Retrieved from https://apps.nsa.gov/iaarchive/ programs /iad-binitiatives/cnsa-

suite.cfm

Sergienko, A. V. (2006). Quantum communications and cryptography (1st ed.). Boca

Raton, FL: Taylor & Francis.

Silverman, J. H. (2009). The arithmetic of elliptic curves (Second ed.). New York:

Springer-Verlag.10.1007/978-0-387-09494-6.

Wang, X., Xu, G., Wang, M., & Meng, X. (2016). Mathematical foundations of public

key cryptography (1st ed.). Boca Raton, FL: CRC.

Washington, L. C. (2003). Elliptic curves: number theory and cryptography. Boca

Raton, FL: Chapman & Hall/CRC.

Xiong, H., Qin, Z., & Vasilakos, A. (2017). Introduction to certificateless cryptography.

Boca Raton, FL: CRC/Taylor & Francis.

Yan, S. Y., & Springer Science+Business Media. (2008). Cryptanalytic attacks on RSA.

New York: Springer.